Finite Group Databasebeta

← #30648729

Subgroups — #30648729

Order 16256.

492 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550424C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic, complemented
3#22550188C212yesmaximal, complemented
4#22550157C211yesmaximal, characteristic, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550008C212yesmaximal, complemented
8D8×Dic127C212yesmaximal, complemented
9#22549956C212yesmaximal, complemented
10#22549928C211yesmaximal, characteristic, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×D4×D127C2211yescharacteristic
14C22×C4×D127—21no—
15C2×#14523233C2212yes—
16#16861209C2211yescharacteristic, complemented
17#16861196—21no—
18#16861180—21no—
19#16861180C2212yes—
20C4×D508C2211yescharacteristic
21D4×Dic127—21no—
22D4×Dic127C2211yescharacteristic
23D4×Dic127C2212yescomplemented
24#16861171C2211yescharacteristic
25C4×C22⋊D127—21no—
26(C22⋊D127)⋊C4—21no—
27C4⋊C4×D127C2211yescharacteristic
28C22⋊C4×D127—21no—
29C42×D127C2211yescharacteristic
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32C4⋊(C127⋊C8)C2211yescharacteristic, complemented
33#16861131C2212yes—
34#16861130C2212yes—
35#16861130C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37C8×Dic127C2212yescomplemented
38C2×D8×C127C2212yes—
39#127×C127C2211yescharacteristic, complemented
40C4×D4×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic, complemented
42#102×C127C2212yescomplemented
43C23×D127—21no—
44#14523247—44no—
45C2×Dic254—21no—
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—22no—
48C2×C22⋊D127—21no—
49D4×D127—22no—
50D4×D127—22no—
51C22×Dic127—21no—
52C22×Dic127—22no—
53C2×C4×D127—21no—
54C2×C4×D127—21no—
55C2×C4×D127—21no—
56C2×C4×D127—44no—
57C2×C4×D127D411yescharacteristic
58C2×C4×D127D411yescharacteristic
59C2×C4×D127C2311yescharacteristic
60#14523233—24no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64C4⋊Dic127C2311yescharacteristic
65Dic127⋊C4—21no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127—21no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—21no—
71C4×Dic127D411yescharacteristic
72C4×Dic127D411yescharacteristic
73C4×Dic127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2312yes—
77C2×D4×C127C2311yescharacteristic
78C22×C4×C127—21no—
79D8×C127—24no—
80C2×C8×C127C2312yes—
81C4⋊C4×C127C2311yescharacteristic
82C22⋊C4×C127—21no—
83C42×C127C2311yescharacteristic
84C22×D127—24no—
85C22×D127—21no—
86C22×D127—42no—
87C22×D127—42no—
88C22×D127C2×D411yescharacteristic
89Dic254—42no—
90D508—21no—
91D508#4112yescomplemented
92C22⋊D127—24no—
93C22⋊D127—44no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127—44no—
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C2×Dic127C2×D411yescharacteristic
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—41no—
104C4×D127—42no—
105C4×D127—42no—
106C4×D127—42no—
107C4×D127D814yescomplemented
108C127⋊C8—22no—
109C23×C127—21no—
110C23×C127—22no—
111D4×C127—21no—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—42no—
115D4×C127—42no—
116D4×C127#4112yescomplemented
117C2×C4×C127—21no—
118C2×C4×C127—42no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2411yescharacteristic
122C8×C127—22no—
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127C22×D411yescharacteristic
130C2×D127—21no—
131C2×D127—21no—
132C2×D127—22no—
133C2×D127—48no—
134C2×D127—41no—
135C2×D127—41no—
136C2×D127C2×D812yes—
137C4×C127—21no—
138C4×C127—41no—
139C4×C127C2×D812yes—
140C4×C127C2×#4111yescharacteristic
141C4×C127C2×#4111yescharacteristic
142Dic127—21no—
143Dic127—21no—
144Dic127—41no—
145Dic127—42no—
146Dic127C2×D812yes—
147C2×C127—42no—
148C2×C127—82no—
149C2×C127#52511yescharacteristic
150C2×C127C22×D811yescharacteristic
151C2×C127#57511yescharacteristic
152D127—22no—
153D127—42no—
154#3100—1271nomaximal, Sylow
155C127#310011yescharacteristic, complemented, Sylow
156#532—1271no—
157C2×C4×D4—1271no—
158#480—1272no—
159#474—1271no—
160#474—1271no—
161#459—1271no—
162#459D12711yescharacteristic, complemented
163#447—1272no—
164C4×D8—1272no—
165C4×D8—1272no—
166#431—1271no—
167C2×#41—2541no—
168C22×D4—1271no—
169C23×C4—2541no—
170C2×D8—1272no—
171C2×D8C2×D12712yes—
172C4⋊Q8—1271no—
173#127C2×D12711yescharacteristic, complemented
174#124—2541no—
175#121—1272no—
176#121—2541no—
177C4×D4—1271no—
178C4×D4—1272no—
179C4×D4—1271no—
180C4×D4—1271no—
181C4×D4—2541no—
182C4×D4—2541no—
183C4×D4—2541no—
184C4×D4C2×D12711yescharacteristic
185C2×C4⋊C4—1271no—
186C2×C22⋊C4—2541no—
187C2×C42—1271no—
188#107—1272no—
189#107—1272no—
190C4⋊C8—1271no—
191C4⋊C8C2×D12711yescharacteristic, complemented
192#102—1272no—
193#102—1272no—
194#102—1272no—
195#102C2×D12712yescomplemented
196C22⋊C8—1272no—
197C4×C8—1272no—
198C24—2541no—
199#41—5084no—
200C2×Q8—2541no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—1271no—
204C2×D4—2541no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4—2542no—
208C2×D4C22×D12712yes—
209C2×D4C22×D12711yescharacteristic
210C22×C4—21no—
211C22×C4—1271no—
212C22×C4—1271no—
213C22×C4—1271no—
214C22×C4—5084no—
215C22×C4—2541no—
216C22×C4—2541no—
217C22×C4—2541no—
218C22×C4—2542no—
219C22×C4—2541no—
220D8—24no—
221D8—2544no—
222C2×C8—1272no—
223C2×C8C22×D12712yes—
224C4⋊C4—1271no—
225C4⋊C4—1272no—
226C4⋊C4—2541no—
227C4⋊C4—2541no—
228C4⋊C4C22×D12711yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—2541no—
231C22⋊C4—2541no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C22⋊C4—2541no—
235C42—1271no—
236C42—1271no—
237C42—1271no—
238C42C22×D12711yescharacteristic
239C23—21no—
240C23—22no—
241C23—1271no—
242C23—5082no—
243C23—5082no—
244C23—2544no—
245C23—2541no—
246Q8—5082no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—1272no—
253D4—5084no—
254D4—2544no—
255D4—2541no—
256D4#1452324712yescomplemented
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—1271no—
260C2×C4—1274no—
261C2×C4—1271no—
262C2×C4—1271no—
263C2×C4—5081no—
264C2×C4—5082no—
265C2×C4—5082no—
266C2×C4—5082no—
267C2×C4—5084no—
268C2×C4—5082no—
269C2×C4—2541no—
270C2×C4—2542no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4D4×D12711yescharacteristic
274C2×C4D4×D12711yescharacteristic
275C2×C4C23×D12711yescharacteristic
276C8—22no—
277C8—2542no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—1272no—
285C22—5088no—
286C22—5081no—
287C22—5081no—
288C22—2542no—
289C22—2541no—
290C22—2541no—
291C22C2×D4×D12711yescharacteristic
292C4—21no—
293C4—41no—
294C4—1272no—
295C4—5081no—
296C4—5082no—
297C4—2541no—
298C4—2541no—
299C4D8×D12712yes—
300C4#41×D12711yescharacteristic
301C4C2×#1452324711yescharacteristic
302C2—42no—
303C2—82no—
304C2—5082no—
305C2—2542no—
306C2#2255038611yescharacteristic
307C2C2×D8×D12711yescharacteristic
308C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3100
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550424C2semidirect
2. C4×D4×D127C2semidirect
3. #22550188C2semidirect
4. #22550157C2semidirect
5. #22550156C2semidirect
6. #22550068C2semidirect
7. #22550008C2semidirect
8. D8×Dic127C2semidirect
9. #22549956C2semidirect
10. #22549928C2semidirect
11. #459×C127C2semidirect
16. #16861209C22semidirect
23. D4×Dic127C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. C4⋊(C127⋊C8)C22semidirect
35. #16861130C22semidirect
37. C8×Dic127C22semidirect
39. #127×C127C22semidirect
41. C4⋊C8×C127C22semidirect
42. #102×C127C22semidirect
91. D508#41semidirect
107. C4×D127D8semidirect
116. D4×C127#41semidirect
155. C127#3100semidirect
162. #459D127semidirect
173. #127C2×D127semidirect
191. C4⋊C8C2×D127semidirect
195. #102C2×D127semidirect
256. D4#14523247semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225504242
C4×D4×D1272
#225501882
#225501572
#225501562
#225500682
#225500082
D8×Dic1272
#225499562
#225499282
#459×C1272
#3100127